Left Hand & Right Hand Limits: Definition, Diagram, Solved Examples & FAQs
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A limit of a function at a point is the value of that function as it approaches a particular input. In other words, we can understand a limit as a border value, never to be reached but infinitesimally close. Limits are used in calculus to define derivatives, and the definite integrals that go along with them can additionally be used to analyze the local behavior of functions near points of interest.
When we talk about the limit of a function around a particular point, we first ask for its existence at that point, and here comes the entry of the left-hand limit and the right-hand limit. The left hand limit (LHL) and right hand limit (RHL) establish the validity of the limit around the point. So the first question arises: "What is this LHL and RHL?" And so the topic of discussion in today's article is.
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In this Maths article, we will be discussing left and right hand limits in detail, and I will be trying to simplify the concept using some real-life examples.
We will also be discussing how non consistency in RHL and LHL determines the continuity of a function.
Left Hand And Right Hand Limits
Let me try to explain the right and left hand limits more simply by using some practical cases. Suppose we are crossing a hanging bridge, and the bridge is made of two sections separated exactly in the middle of the bridge. Now, assume that one side of the bridge is at a high altitude and the other side is low. Now tell me, will you be able to cross the bridge? you won’t. Right? simply because your bridge is discontinuous. Similarly, if the left-hand limit, which we call LHL, and the right-hand limit, or RHL, of a function are unequal, that means your function is discontinuous. And that is simply the reason why the limit does not exist because your function is not continuous, or you can simply say it is discontinuous. In short, for a limit to be defined, we must have LHL=RHL. I hope you got how LHL and RHL decide the continuity of a function. Now, let's define what this RHL and LHL exactly means. We will be discussing the definition in the next section.
So basically, LHL and RHL describe the behaviour of the function at the immediate left and immediate right of the input value respectively. Not clear?
For example, check this graph.
In (a), the RHL of the function is defined, because the graph approaches a fixed value from the right. The LHL is undefined because the graph is not approaching a fixed value. Similarly, in (b), the RHL is also undefined, and the LHL is defined. And finally, in (c), both the RHL and LHL are defined, but they aren't consistent. Hence, in all the cases, the limit is undefined.
I hope you are now well versed with the graphical interpretation of RHL, LHL and limits. Now, let's solve a problem on how to evaluate a given limit mathematically.
Right and Left Hand Limits Definition
A point to notice here is that, we are not talking about the value of the function at x=a. We are rather concerned with the value that function tends towards as x approaches the value x=a from the right hand side or left hand side of the point x=a.
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How to find Right and Left Hand Limits
Follow the algorithm to find Right hand limit and the left hand limit.
For Left hand limit :
For Left hand limit :
Step 1: Write
Step 2: Now, put
Step 3: Simplify
Step 4: The value obtained in step 3 is the Left hand limit of
Similarly, for Left hand limit:
Step 1: Write
Step 2: Now, put
Step 3: Simplify
Step 4: The value obtained in step 3 is the Right hand limit of
Let's do an example, suppose,
Solution: Clearly we know that the first step to evaluating LHL and RHL is to just put the value around which the limit needs to be calculated in the function.
Similarly,
\(\because LHL = RHL = 3, the limit exists, such that,
=> \displaystyle\lim_{x\to 1^}f(x) = 1\)
For more problems on evaluation of the left hand and right hand limit you can move to the solved examples section.
Understanding Three Important Ideas About a Function f(x)
When we look at a function f(x), there are three key things we often talk about when xxx is getting close to a certain value a:
- Left-hand limit (LHL)
- Notation: lim (as x → a⁻) f(x)
- Meaning: Describes how f(x) behaves when x gets close to a from the left side (values smaller than a).
- Right-hand limit (RHL)
- Notation: lim (as x → a⁺) f(x)
- Meaning: Describes how f(x) behaves when x gets close to a from the right side (values larger than a).
- Notation: lim (as x → a⁺) f(x)
- Actual value of the function at x = a
- Notation: f(a)
- Meaning: This is the exact value of the function when x equals a.
- Notation: f(a)
Properties of Left Hand & Right Hand Limits1. Existence of a Limit
The existence of a limit at a particular point in a function depends on the behavior of the function as it approaches that point from both sides. If the function tends toward the same value from the left and from the right, the limit is said to exist at that point. This concept ensures a predictable and consistent approach toward the point in question.
2. Different Left and Right Limits
When the values that a function approaches from the left and right of a point are not the same, the function exhibits a discontinuity at that point, and the limit is said not to exist. This scenario often indicates a sudden jump or gap in the graph of the function, reflecting a change in behavior on either side of the point.
3. Continuity Check
A function is considered continuous at a point if it is smoothly connected there, with no breaks, jumps, or holes. For continuity to hold, the function must be defined at the point, and its behavior from both sides must align with its actual value at that point. This ensures the graph of the function flows without interruption.
4. Used for Piecewise Functions
In the case of piecewise-defined functions — functions described by different expressions over various intervals — examining the behavior on both sides of the joining points is essential. Left-hand and right-hand limits help determine whether the function transitions smoothly from one piece to the next, or whether there is a break or mismatch at the boundary.
Applications of Left Hand & Right Hand Limits
1.Checking Continuity
LHL and RHL are used to test whether a function is smooth (continuous) or has a jump or hole.
2.Understanding Graph Behavior
They help in analyzing the behavior of graphs near corners, edges, or breaks.
3.Solving Real-World Problems
In physics and engineering, they help in studying sudden changes, like shock waves or switching signals.
4.Limits in Calculus
These limits are the foundation for studying derivatives and integrals later in calculus.
How to find Right and Left Hand LimitsSummary
Let's summarize whatever we have discussed so far in short so that you can prepare your short notes for quick and effective revisions:
- A limit of a function at a point is the value of that function as it approaches a particular input.
- LHL is the value to which the function approaches when it approaches the point from the left and the RHL is the value to which the function approaches when it approaches the point from the right.
- Basically, LHL and RHL describe the behavior of the function at the immediate left and immediate right of the input value respectively.
- Left hand limit and right hand limit are very important as it tells us if the limit of the function exists at a particular point. Additionally it also tells us whether the function is continuous or not.
- For a limit to be defined, we must have LHL=RHL.
Solved example
Example 1. A function is defined by,
Find the value of k, for which
Solution 1. Clearly, we know that, for
Similarly, we have,
From, the definition of limit, we have,
\(=> 2k + 7 = 23 –––––––(from (1) and (2))
=> 2k = 16
=> k = 8\)
Hence, the required value of k for which the limit is defined is 8.
Example 2. Consider a function f(x), whose graph is depicted as follows:
Evaluate the limit
Solution 2. From the graph, we obtained,
\(=> LHL = \displaystyle\lim_{x\to3^–}f(x) = 1, &
=> RHL = \displaystyle\lim_{x\to3^+}f(x) = 4\)
Clearly, from the definition of limit, for a limit to be defined, we must have,
Hence,
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FAQ For Left hand right hand limits
How do you evaluate left hand and right hand limits?
The first step to evaluating LHL and RHL is to just put the value around which the limit needs to be calculated in the function. If it works, well and good; otherwise, we will be applying the properties of limits.
What happens if the left hand limit is not equal to the right hand limit?
If the left hand limit is not equal to the right hand limit, the limit of the function doesn't exist and the function might oscillate between the two values.
Why do we have a left hand limit and a right hand limit?
Left hand limit and right hand limit are very important as it tells us if the limit of the function exists at a particular point. Additionally it also tells us whether the function is continuous or not.
Can the right hand limit not exist?
Yes, it can be that the right-hand limit of a function doesn't exist. In that case, the limit of the function at point does not exist.
What is the symbolic notation of the left hand limit?
Left hand limit or LHL of a function can be symbolically represented as,
What if left and right limits are not equal?
If the left-hand limit and right-hand limit are not equal, then the limit does not exist at that point.
Are left and right hand limits used in real life?
Yes! They help understand sudden changes in motion, economics, and piecewise functions where a rule changes before or after a point.
What is the main use of left and right-hand limits?
A: Left and right-hand limits help determine the behavior of a function near a point, especially for piecewise functions or where a function has a sudden change.
Can a function be discontinuous even if both limits exist?
Yes. A function can be discontinuous at a point even if both left and right-hand limits exist but do not equal the value of the function at that point.
How do I know which piece of a piecewise function to use for LHL or RHL?
Use the “less than” part (e.g., xa) for the right-hand limit.